Abelian groups


Definition

Groups + xy = yx

Examples

Structure

Every finitely generated abelian group is isomorphic to some
              Z_{q_1} x Z_{q_2} x ... x Z_{q_n} x Z^r
        
where q_1 divides q_2 divides ... divides q_n and r >= 0. r is called the rank of the abelian group, and q_i the invariants. The rank and invariants are uniquely determined by the group. For example, there are exactly six nonisomorphic abelian groups of order 360 = 2^3 . 3^2 . 5, namely
             Z_360
             Z_2 x Z_180
             Z_2 x Z_2 x Z_90
             Z_2 x Z_6 x Z_30
             Z_3 x Z_120
             Z_6 x Z_60
        

Representation

Decision problems

Identity problem: Solvable
Word problem: Solvable

Spectra and growth

Finite spectrum: 1,1,1,2,1,1,1,3,2,1,...
Free spectrum: 1, aleph_null, aleph_null, ...
Growth series: ((1+z)/(1-z))^r for the free abelian group on r generators

History/Importance

References

[Kurosh]

Subsystems